Why should we bother to think about writing numbers? You’ve probably been writing numbers, without any problems, since you were about 5 years old.
There are two reasons. The first is that writing very large numbers like 1 000 000 000 000 is a pain – there must be a better way of doing it. And in Post 16.6 we met an even larger number. The second is that many of our ideas (Posts 16.3 and 16.6) are based on how precisely we can measure something. It would be useful to be able to write numbers in a way that makes this precision clear.
The number 1 000 000 000 000 is equal to which is 10 multiplied by itself 12 times; this is why it is written as 1 followed by twelve zeros. We can write this number much more compactly as 1012. We can extend this idea. So, 3 000 = 3 × 1 000 = 3 × 103. Similarly, 41 250 = 4.125 × 10 000 = 4.125 × 104.
We can also extend this method to numbers that are smaller than 1, as shown in the table below.
You will notice that the superscript number (1, 2, 3…) is always equal to the number of zeros after the decimal point minus 1. So 0.0013 = 1.3 × 0.001 = 1.3 ×10-3.
Suppose we have measured something with an ordinary ruler and found that is 10 cm long. We can easily distinguish this length from 9.9 cm and 10.1 cm. To make this clear, we report our measurement as 10.0 cm.
When we calculate something from experimental measurements, the result is only as precise as the least precise measurement. Suppose your car indicates that you have just finished a journey of 3.5 km and your watch tells you that it lasted 4.05 minutes. Then your average speed is 3.5 ÷ (4.05/60) = 3.5 × (60/4.05) = 51.85185 km per hour. But you only know the distance (3.5 km) with a two-digit precision. So you only know the speed with that precision. You may think that it should then be reported as 51 km per hour. But the answer you got from your calculations is nearer to 52 than 51, so your calculated average speed is 52 km per hour.
Supposing on another trip, you calculated an average speed of 52.50000 km per hour, using the same measurement methods. What is your best estimate of the speed? Your answer is mid-way between 52 and 53; by convention, you report your speed as 53 km per hour. The reason for this convention is that your calculator may only display answers to 5 decimal places. Any further non-zero digits after the fifth decimal place, in this example, would make the result of your calculation closer to 53 than 52.
I know of an engineering textbook aimed at first year university students that often gets this wrong. So writing numbers isn’t as simple as it might appear!